Thanks for the link. They have a .pdf of chapter one which I gave a look. Right there on page 14 when estimating the mass of our galaxy inside our suns radius we have:
"The approximation that the mass distribution is spherical is reasonable for the dark halo, but not for the flat stellar disk. Better models suggest that this estimate is probably high by about 30%, since a disk requires less mass to produce a given centripetal acceleration"
And there you have it. While he notes that a spherical distribution is different than a flat disk, he estimates the error at about 30%. The problem is that the error is not one of a constant factor, it's a different function. This unspecified better model also doesn't seem to care about the matter outside the solar radius - which also affects our velocity in a disk, but not in a sphere.
There is a lot of math in chapter one, but also a lot of hand waving.